Asymmetry of convex polytopes and vertex index of symmetric convex bodies
type: Item
instance of: scholarly article
title: Asymmetry of convex polytopes and vertex index of symmetric convex bodies
zbMATH Open document ID: 1161.52008
published in: Discrete \& Computational Geometry
publication date: 2009-03-24T00:00:00Z
review text: In [Geom. Dedicata 90, 45--48 (2002; Zbl 0999.52008)] the authors have shown that a~polytope in~\({\mathbb R}^n\) with few vertices is far from being symmetric with respect to the Banach--Mazur distance. More precisely, it was established that Banach--Mazur distance between such a~polytope and any symmetric convex body is large. In this paper they introduce a~new, averaging-type parameter, to measure the asymmetry of polytopes. It turns out that, surprisingly, this new parameter is still very large, and in fact it satisfies the same lower bound as the Banach--Mazur distance. In a~sense it exhibits the following phenomenon: if a convex polytope with small number of vertices is as close to a~symmetric body as it can be, then most of its vertices are as~bad as the worst one. The authors apply their results to provide a lower estimate on the vertex index of a~symmetric convex body, which was recently introduced in [Adv. Math. 215, No. 2, 626--641 (2007; Zbl 1131.46011)] by \textit{K.~Bezdek} and \textit{A.~E.~Litvak}. Furthermore, the affirmative answer is given to a~conjecture by \textit{K.~Bezdek} [Problem 6.3 of Period. Math. Hung. 53, 59--69 (2006; Zbl 1127.52026)] on the quantitative illumination problem.
Mathematics Subject Classification ID: 52B11
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52B15
Mathematics Subject Classification ID: 51M16
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 46B07
zbMATH DE Number: 5532462
zbMATH Keywords: asymmetry of polytopes
zbMATH Keywords: illumination parameter
zbMATH Keywords: measures of symmetry
zbMATH Keywords: polytopes with few vertices
zbMATH Keywords: vertex index
author: Efim D. Gluskin
author: Alexander E. Litvak
reviewed by: A. R. Alimov
MaRDI profile type: Publication
full work available at URL: s00454-008-9075-9
OpenAlex ID: W2064896804
cites work: Q3138750
cites work: The illumination conjecture and its extensions
cites work: On the vertex index of convex bodies
cites work: On the asymmetry constant of a body with few vertices
cites work: Q5571194
cites work: Q5517554
cites work: Combinatorial problems on the illumination of convex bodies
cites work: A Sufficient Condition for an Integral Operator to Have a Trace.
cites work: Inequalities between the Two Kinds of Eigenvalues of a Linear Transformation
DOI: 10.1007/S00454-008-9075-9
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description: scientific article; zbMATH DE number 5532462